# Compute initial value in Kalman Smoother

Suppose we observe data $$y_t$$ and $$X_t$$ from $$t=1,...,T$$ and want to estimate a dynamic linear model of the form

$$y_t = X_t\beta_t + \epsilon_t$$

$$\beta_t = \beta_{t-1} + \omega_t$$

where $$\epsilon_t$$ and $$\omega_t$$ are normally distributed with zero mean and known variances. Suppose that the initial value $$\beta_0$$ follows a normal distribution with known moments s.t. $$\beta_0 \sim N(a_0,P_0)$$.

The state sequence $$\beta_1,...,\beta_T$$ can easily be estimated using the Kalman filter and Kalman smoother that are for instance outlined in the book of Durbin & Koopman. They provide the necessary forward filtering and backward smoothing recursions for the time periods $$1,...,T$$.

Usually, one would initialize the forward filtering procedure by setting the filtered mean to $$a_0$$ and the filtered variance to $$P_0$$ in period $$t=0$$. This initialization then allows us to forward filter from $$t=1$$ up to $$t=T$$.

Then, the Kalman smoother updates the filtered state means and variances using backward recursions, going from $$t=T$$ to $$t=1$$. All of this works fine and the estimates are looking good. However, I would like to also update $$a_0$$ and $$P_0$$ in the backward smoothing process. How can we do that? If I follow the usual backward recursions, it appears that I would need data in $$t=0$$ to do so.

Edit:

I think the question is equivalent to:

Are the filtered estimates at $$t=T$$ equvialent to the smoothed estimates at $$t=T$$ and do we therefore have to do the backward recursions from $$t-1$$ to $$0$$ after initializing the smoothed estimates at $$T$$ using the filtered estimates at $$T$$?